Integrated circuits manufactured in a silicon process, when operating in high-frequency ranges (e.g. 300 MHz to 20 GHz), have significant high-frequency effects among the passive structures. The passive structures include spiral inductors made from metal conductors and interconnects between devices. Intentional devices, such as inductors and transformers, are modeled through parameter extraction, in which the essential circuit parameters are obtained through complex steps of analyzing the underlying physics. Unintentional structures, such as interconnects that link devices together for certain functions, are modeled through parasitic extraction, in which the self-terms or mutual-terms are computed through the similar complex steps of analyzing the underlying physics.
At high-frequency ranges, the distinction of parameter extraction and parasitic extraction is blurred, due to the strong couplings between passive devices and passive interconnects. In order to account for the total effects, a versatile yet efficient extraction or simulation tool for passive structures is necessary. Efficiency of the tool, preferably an interactive one, is essential for the design productivity.
It is well known that electromagnetics is the underlying physics that govern passive devices, yet the traditional full-wave implementation is extremely slow and often running from hours to days for a simple extraction problem. One widely used method in solving the electromagnetic equations in a layered media is the Mixed-Potential Integral Equation (MPIE) method, described by S. M. Rao, et al, in IEEE Transactions on Antennas and Propagation, 30(3):409-418, May, 1982, and entitled “Electromagnetic Scattering by Surfaces of Arbitrary Shape.”
Traditionally, this method is implemented in a 2.5 D simulation context where the conductor is assumed to be infinitely thin. This assumption is reasonably valid in microwave circuits where the width/thickness ratio is relatively large. However, such assumption becomes increasingly invalid for integrated circuits where the conductor thickness can be on the same order of the width. One serious defect in the popular rooftop triangular current basis function is the high matrix condition number at low frequencies, making the method unstable for broadband modeling applications. The curl-free and divergence-free decomposition method reduced the condition number at a cost of high implementation difficulty and disassociation with the circuit concepts. The method is discussed in S. Kapur, et al. in U.S. Pat. No. 6,513,001, entitled Efficient Electromagnetic Full-Wave Simulation In Layered Semiconductor Media.
Another widely popular theoretical approach is the Partial Element Equivalent Circuit (PEEC) of A. E. Ruehli, discussed in IEEE Transactions on Microwave Theory and Techniques, 40(7):1507-1516, July 1992 and entitled “Circuit models for three-dimentional geometris including dielectrics.” (Note that references to PEEC herein are general references to partial-element-equivalent-circuit methods and is not intended to be limited to one particular type of PEEC method.) PEEC method discretizes the MPIE problem into an equivalent circuit of vast number of partial elements. Even though in theory PEEC is an integral equation solving implementation that closely relates the electromagnetic problem to a circuit approximation, its practical use is very limited in high-frequency IC passive structure modeling due to prohibitively high computing cost needed to arrive at a reasonable solution. Partial capacitances and partial inductances are computed by methods such as those proposed by K. Nabors, et al, in a publication entitled A Multipole Accelerated 3-D Capacitance Extraction Program published in IEEE Transactions On Computer-Aided Design Of Integrated Circuits And Systems, 10(11): 1447-59, November 1991 and by M. Kamon, et al, in a publication entitled A Multipole-Accelerated 3-D Inductance Extraction Program published in IEEE Transactions On Microwave Theory And Techniques, 42(9): 1750-8, September 1994.
Yet it is further understood that as frequency increases, the retardation of the electromagnetic influence must be considered to account for the time delay in the electromagnetic coupling because of vastly complex PEEC elements. This would further limit the applicability of PEEC in high-frequency circuit modeling. A need thus arises to incorporate certain electromagnetic behaviors directly into a circuit simulation context. Jandhvala, et al. in a publication entitled A Surface Based Integral Equation Formulation for Coupled Electromagnetic and Circuit Simulation published in Microwave Optical Technology Letters, Vol 34, No. 2, pp. 102-106, Jul. 20, 2002, attempt to use a slightly more high-order discretization scheme, the surface-based integral equation formulation, to link the circuit simulation with electromagnetic simulation, yet its use of the free-space Green's function causes it to suffer from the same limited usefulness as the traditional PEEC method.
Accordingly, what is needed is a method and apparatus that reduces the electromagnetic simulation in layered media into a similar PEEC discretization without the complexity of the full 3D discretization of dielectric and substrate as required by the traditional PEEC method, and yet fully accounts for the electromagnetic effects including the substrate eddy current loss and displacement loss. Moreover, it should closely relate to the circuit concepts such as voltage, current, and affords an easy way of integrating with traditional circuit simulators.